Gravitational Effective Theories with Maximal Supersymmetry and a Peculiar Parity

We study the space of four-dimensional ultraviolet completions for $\mathcal{N}=8$ supergravity that are described at low energies by weakly-coupled effective field theories (EFTs) with maximal supersymmetry and $\mathrm{SU}(4)\times\mathrm{SU}(4)$ R-symmetry. We show that tree-level factorization of the 4-, 5-, and 6-point EFT scattering amplitudes, together with a certain ``peculiar parity'' condition, leads to nonlinear constraints on the 4-point Wilson coefficients. This peculiar parity is a property that can only be imposed on a subset of scalar amplitudes. Combining the nonlinear constraints with positivity, we find that the allowed region of 4-point Wilson coefficients is reduced to a non-convex domain with two sharp corners: one being the closed superstring Virasoro--Shapiro amplitude, the other an infinite spin tower amplitude exchanging states of every spin at the same mass. We show both numerically and analytically that requiring a finite number of states near the first mass level leaves only the Virasoro--Shapiro amplitude.